7. The random variable \(X\), which can take any value from \(\{ 1,2 , \ldots , n \}\), is modelled by the discrete uniform distribution with mean 10 .
- Show that \(n = 19\) and find the variance of \(X\).
- Find \(\mathrm { P } ( 3 < X \leq 6 )\).
The random variable \(Y\) is defined by \(Y = 3 ( X - 10 )\).
- State the mean and the variance of \(Y\).
The model for the distribution of \(X\) is found to be unsatisfactory, and in a refined model the probability distribution of \(X\) is taken to be
$$\mathrm { f } ( x ) = \left\{ \begin{array} { c l }
k ( x + 1 ) & x = 1,2 , \ldots , 19
0 & \text { otherwise }
\end{array} \right.$$ - Show that \(k = \frac { 1 } { 209 }\).
- Find \(\mathrm { P } ( 3 < X \leq 6 )\) using this model.